Local density of states in metal-topological superconductor hybrid systems
Marco Gibertini,1,* Fabio Taddei,2 Marco Polini,2 and Rosario Fazio1,†1NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy2NEST, Istituto Nanoscienze-CNR and Scuola Normale Superiore, I-56126 Pisa, Italy
(Received 29 November 2011; revised manuscript received 24 February 2012; published 30 April 2012)
We study by means of the recursive Green’s function technique the local density of states of (finite andsemi-infinite) multiband spin-orbit-coupled semiconducting nanowires in proximity to an s-wave superconductorand attached to normal-metal electrodes. When the nanowire is coupled to a normal electrode, the zero-energypeak, corresponding to the Majorana state in the topological phase, broadens with increasing transmission betweenthe wire and the leads, eventually disappearing for ideal interfaces. Interestingly, for a finite transmission a peakis present also in the normal electrode, even though it has a smaller amplitude and broadens more rapidly withthe strength of the coupling. Unpaired Majorana states can survive close to a topological phase transition evenwhen the number of open channels (defined in the absence of superconductivity) is even. We finally studythe Andreev-bound-state spectrum in superconductor-normal metal-superconductor junctions and find that inmultiband nanowires the distinction between topologically trivial and nontrivial systems based on the number ofzero-energy crossings is preserved.
Since the first prediction1 of real solutions to the Diracequation, known as Majorana fermions, there have beenmany attempts to demonstrate their occurrence in nature,2
but clear evidence is still lacking. Besides the natural searchfor these elusive particles in high-energy physics, it hasbeen recently suggested that Majorana fermions can existas exotic excitations in certain condensed-matter systems.3,4
Such solid-state realizations include fractional quantum Hallstates at filling factor ν = 5/25, p-wave superconductorsand superfluids,6,7 three-dimensional topological insulators inproximity to s-wave superconductors,8 as well as spin-orbitcoupled semiconductors in a magnetic field with proximity-induced s-wave superconducting pairing.9–14 The importanceof finding Majorana fermions in condensed-matter systems isnot only related to their fundamental interests. It is also rootedin the non-Abelian braiding statistics of these particles, whichcould be exploited as a basis for decoherence-free topologicalquantum computation.15
In this paper we focus on a specific proposal realized withspin-orbit-coupled semiconducting nanowires in proximityto an s-wave superconductor (S) and subjected to an in-plane magnetic field12–14 (a system that, for the sake ofsimplicity, is henceforth termed “S-nanowire”). This wirecan support Majorana-fermion bound states at its ends whenparameters such as chemical potential, magnetic field, andsuperconducting pairing are properly tuned.16 The S-nanowireis said to be in the topological phase when Majorana boundstates are present, while it is topologically trivial otherwise.
In order to assess the presence of Majorana fermionsin such solid-state systems, it is of primary importanceto predict clear signatures of the topological phase, whichcould be then used to guide experiments. A very relevantquantity is the local density of states (LDOS), which canbe accessed in scanning tunneling microscopy. LDOS cal-culations have already been carried out in Refs. 17 and18, but were restricted to finite-size S-nanowires in whichsuperconductivity is uniform along the wire. The influence
of normal segments inside a one-dimensional S-nanowirewas investigated in Ref. 19. In this work we exploit therecursive Green’s function technique to study the LDOS ofboth finite and semi-infinite multiband S-nanowires in thepresence of normal electrodes (NS junctions), as well assupercoductor-normal metal-superconductor (SNS) junctions.In this manner we have been able to (i) reproduce previousresults,17–19 (ii) generalize them to account for the couplingto normal electrodes and/or for a finite width of the wire, and(iii) consider more complicated structures like SNS junctions.
The paper is organized as follows. In Sec. II we describe themodel and we introduce the corresponding Hamiltonian. Thephase diagram is presented and new analytical expressions forthe phase boundaries are reported. Section III is devoted tothe discussion of our numerical results for the LDOS. Westart in Sec. III A with the case of a finite-size multibandS-nanowire.17 We then consider a single NS junction betweensemi-infinite leads in Sec. III B. This situation can arise forinstance when the nanowire is only partially in proximityto a bulk superconductor so that part of the nanowire is inthe normal state. Here we also study the evolution of theLDOS across the topologically trivial/topologically nontrivialphase transition. In Sec. III C we turn our attention to a SNSjunction, illustrating numerical results for the LDOS and forthe Andreev-bound-state spectrum. Finally, in Sec. IV we drawour main conclusions.
II. MODEL HAMILTONIAN
Our calculation of the LDOS is based on a recursiveGreen’s function method especially designed for tight-bindingHamiltonians. In order to apply this method to the presentcase, we describe the semiconducting nanowire as a squarelattice with a finite width W in the direction perpendicularto the axis of the wire (the y direction) and lattice constanta. Assuming the presence of Rashba-type spin-orbit (SO)coupling, of strength α, and a Zeeman field V along thewire (x direction), the lattice discretization of the usual
GIBERTINI, TADDEI, POLINI, AND FAZIO PHYSICAL REVIEW B 85, 144525 (2012)
continuum-model Hamiltonian13,14,20 reads16
HRZ =∑i,j
[hRZ(i,j )]σσ ′ c†i,σ cj,σ ′
= −t∑
〈i,j〉,σc†i,σ cj,σ + (ε0 − μ)
∑i,σ
c†i,σ ci,σ
+ iα∑
〈i,j〉,σ,σ ′
(ν ′
ij σxσσ ′ − νijσ
y
σσ ′)c†i,σ cj,σ ′
+V∑i,σ,σ ′
σxσσ ′ c
†i,σ ci,σ ′ . (1)
Here ε0 = 4t is a uniform on-site energy which sets the zeroof energy, σ i are spin- 1
2 Pauli matrices, νij = x · dij and ν ′ij =
y · dij , with dij = (r i − rj )/|r i − rj | being the unit vectorconnecting site j to site i.
If we now allow sections of the nanowire to be in contactwith a bulk s-wave superconductor, the proximity effectinduces a nonvanishing superconducting pairing in thesesections so that the complete Hamiltonian becomes
H = HRZ + HS, (2)
where
HS =∑
i
[�(i)c†i,↑c†i,↓ + H.c.]. (3)
For simplicity we assume �(i) to be piecewise constant, with|�(i)| = � in the regions in contact with the superconductorand �(i) = 0 otherwise. Moreover, we assume that a barrier ispresent at the boundary between proximized and nonprox-imized sections, leading to a decrease of the value of thehopping energy t and of the SO coupling α by the same factorγ .21
Finally, it is convenient to introduce the Nambu spinors�i = (ci,↑,ci,↓,c
†i,↓, − c
†i,↑)T and rewrite the Hamiltonian (2)
in the form
H = 1
2
∑i,j
�†i HBdG(i,j )�j , (4)
where
HBdG(i,j ) =(
hRZ(i,j ) �(i)δij
�(i)δij −σyh∗RZ(i,j )σy
)(5)
is the Bogoliubov-de Gennes (BdG) Hamiltonian.22
The tight-binding Hamiltonian (5) will be the starting pointof our analysis.
A. Phase diagram
Before proceeding with the study of the LDOS we needto know in which regions of parameter space we shouldexpect Majorana fermions. This problem has been addressed inRefs. 13 and 14 in the strictly one-dimensional case and in thecontinuum limit (a → 0): The S-nanowire is in the topologicalphase when |V | >
√μ2 + �2, while it is in the trivial phase
otherwise. Thus, the phase boundary occurs along the lineimplicitly defined by
V 2 = μ2 + �2. (6)
0.0 0.5 1.0 1.5 2.0 2.5 3.0μ/t
0.0
0.2
0.4
0.6
0.8
1.0
V/t
FIG. 1. (Color online) Phase diagram of an infinite superconduct-ing wire as a function of the chemical potential μ and the Zeemanfield V . White regions correspond to a trivial system (Q = +1), whiledark regions identify the topologically nontrivial phase (Q = −1).Red thick lines illustrate the prediction in Eq. (8) for the phaseboundaries. These results refer to the following set of parameters:W/a = 10, α/t = 0.1, and �/t = 0.1.
The phase diagram of multiband (W/a = 1) nanowires hasbeen investigated numerically in Refs. 17 and 20. The phaseof a uniform system can be determined, for instance, fromthe evaluation of the following Pfaffian formula23 for thetopological invariant
Q = sign{Pf[HBdG(0)σyτ y]Pf[HBdG(π/a)σyτ y]}, (7)
where HBdG(kx) is the Fourier transform of the BdG Hamil-tonian in Eq. (5), while τ i are Pauli matrices acting onthe particle-hole degrees of freedom. In Fig. 1 we reportnumerical results for a given system (W/a = 10, α/t = 0.1,and �/t = 0.1) as a function of the chemical potential μ
and Zeeman field V , obtained using an algorithm developedby Wimmer.24 The topologically trivial phase corresponds toQ = +1 (white regions), while the nontrivial one is signaledby Q = −1 (dark regions). The phase boundaries in this figure(red thick lines) have been derived analytically and are givenby the following result:
(μ − ε0 − ελ ± 2t)2 + �2 = V 2. (8)
Here ε0 + ελ ∓ 2t are the eigenenergies of HBdG(k) for k =0,π/a, respectively, when V = � = μ = 0. The followingexpression holds for the energies ελ
ελ = −2√
t2 + α2 cos
(λπ
n + 1
); λ = 1, . . . ,n = W/a.
(9)
A thorough derivation of Eqs. (8) and (9) is given inAppendix A.
III. LDOS: NUMERICAL RESULTS
As we mentioned in the Introduction, the LDOS can beaccessed in experiments using scanning tunneling microscopy(STM). The STM experimental setup is sketched in Fig. 2.When the metallic tip of the STM is moved close to thenanowire a tunneling current can flow. By locally measuringthis current I as a function of the tip-sample bias voltage V , the
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STM
N-nanowire S-nanowire
V
Insulator Superconductor
FIG. 2. (Color online) Experimental setup adopted in measure-ments of the LDOS of a nanowire (in this case comprising a normalelectrode and a superconducting segment separated by a barrier). Thedifferential conductance obtained from the current between the tipof the scanning tunneling microscope (STM) and the nanowire isproportional to the LDOS of the nanowire according to Eq. (10).
LDOS at a given position r and energy E can be reconstructedfrom the differential conductance at eV = E:25,26
dI
dV(r,eV ) ∝ N (r,E = eV ). (10)
As a consequence, it is particularly interesting to investigatethe LDOS theoretically and make predictions that can be testedin experiments.
In what follows the LDOS is computed through the standardrelation
N (r,E) = − 1
2π�m{Tr[G(r,E)]}, (11)
where G(r,E) is the Green’s function and the factor 2 inthe denominator is introduced to avoid a double counting ofparticle and hole degrees-of-freedom intrinsic in the BdGformalism. We have computed G(r,E) using a recursiveGreen’s function technique similar to the one adopted inRef. 17, suitably generalized to include the effects of semi-infinite leads.27 For simplicity, in the following we fix thewidth W = 10a, the SO coupling strength α = 0.1t , and thesuperconducting pairing � = 0.1t .
A. Isolated S-nanowire
Let us first consider an isolated S-nanowire of finite length(L = 100a). This situation has been addressed before17,18 andit is considered here for the sake of reference. We considertwo cases: (i) μ = 0 and V/t = 0.2 (with one open channelin the absence of superconducting pairing) and (ii) μ = 0 andV/t = 0.6 (with two open channels). According to Fig. 1,in case (i) the wire is topologically nontrivial, while in case(ii) the wire is topologically trivial. For case (i), Fig. 3(a) showsthat the LDOS at an energy close to the chemical potential ischaracterized by the presence of bound states at both endsof the wire. The presence of Majorana bound states appearsas a sharp peak at zero energy in the LDOS as a function ofenergy at a given position in space [see Fig. 3(b)]. Accordingto Fig. 3(c), these Majorana bound states have oscillating wavefunctions which decay exponentially inside the bulk of the S-nanowire with a typical length scale (effective superconductingcoherence length) ξ ≈ 10a.
(a)
(b) (c)
FIG. 3. (Color online) (a) LDOS of an isolated superconductingnanowire in the topologically nontrivial phase (μ/t = 0, V/t = 0.2)at an energy very close to the chemical potential (E � 0). Boundstates at both ends of the wire are apparent. (b) LDOS at a givenposition (x/a = 4, y/a = 5) as a function of energy. A sharp peakcorresponding to a Majorana bound state is present at E = 0.(c) LDOS at E � 0 as a function of x along the middle of the wire(y/a = 5).
On the contrary, in case (ii) where the wire is topologicallytrivial (and presents two transverse channels in the absence ofsuperconducting pairing) the LDOS at the chemical-potentialenergy is almost zero throughout the wire, while it showsspatial features only at finite energies [see Fig. 4(a) for E/t �±0.002, where E is measured from the chemical potential].In particular, as shown in Fig. 4(c), the LDOS oscillates anddecreases moving toward the center of the wire, the lengthscale of the exponential drop, ξ ≈ 30a, being much larger
(b) (c)
(a)
FIG. 4. (Color online) (a) LDOS of an isolated superconductingnanowire in the topologically trivial phase (μ/t = 0, V/t = 0.6)at E/t � 0.002. (b) LDOS at a given position (x/a = 2, y/a = 5)as a function of energy. Owing to the coupling between the twoMajorana end states, two Dirac-fermion modes appear at finite energy(E/t � 0.002). (c) LDOS at E/t � 0.002 as a function of x alongthe middle of the wire (y/a = 5).
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than for case (i). Now, if the two channels were decoupled, twoMajorana modes would have appeared at each end of the wire,one for each open channel. However, since the two transversechannels are actually coupled in the wire, a single fermionat each end appears at a finite energy,17 as though comingfrom the hybridization of the two Majorana modes. Figure4(b) shows one pair of peaks at E/t � ±0.002 and anotherpair (with smaller amplitude) at E/t � ±0.007. The presenceof two pairs of peaks is due to the long coherence lengthwhich allows the two localized fermions at the ends of thewire to strongly hybridize, lifting the parity degeneracy of thesystem. We have checked that for a longer wire (L/a = 400)the overlap between the fermions vanishes and the two peaksat E/t � 0.002 and E/t � 0.007 merge into a single double-degenerate peak at energy E/t � 0.001. The latter energydepends on the width of the nanowire.28
B. S-nanowire attached to a normal electrode
In this section we analyze the impact of a normal leadattached to the S-nanowire. This situation can arise, forinstance, when the nanowire is only partially in proximityto a bulk superconductor so that part of the nanowire is in thenormal state (as shown in Fig. 2). In order to get rid of finite-size effects, we consider a semi-infinite S-nanowire coupled toa normal lead. The case of a finite-length S-nanowire coupledto two normal leads at both ends does not yield additionalsignificant information.
In Fig. 5 we plot the LDOS, at different positions, as afunction of energy for several values of the barrier strength
FIG. 5. LDOS of a superconducting nanowire coupled to a normallead as a function of energy, at fixed positions in space, in thenontrivial [μ/t = 0 and V/t = 0.2, panels (a) and (c)], and trivial[μ/t = 0, V/t = 0.6, panels (b) and (d)] phase. Panels (a) and (b)refer to a position just inside the superconducting part (x/a = 51),while panels (c) and (d) refer to a position just inside the normal partof the junction (x/a = 50). The interface is at x/a = 50.5.
FIG. 6. (Color online) Fitted values [through Eq. (12)] of the halfwidth at half maximum �/2 (blue circles) and height at zero energyN (red triangles) of the peak in the LDOS at a position just insidethe superconducting nanowire in the nontrivial phase. Solid lines arejust guides for the eye.
γ . The two plots on the left [(a) and (c)] are for a nontrivialnanowire with μ/t = 0 and V/t = 0.2, while the two plots onthe right [(b) and (d)] refer to a topologically trivial nanowirewith μ/t = 0 and V/t = 0.6. Moreover, the top panels [(a)and (b)] refer to a position close to the interface in the S-nanowire, while the bottom panels [(c) and (d)] to a positionclose to the interface in the normal lead. When γ is small theLDOS in the S-nanowire presents a finite gap Eg which isjust a fraction of the superconducting pairing � owing to thepresence of the Zeeman field. Namely, Eg � 0.03t = 0.3�
for the nontrivial nanowire and Eg � 0.018t = 0.18� for thetrivial case. Within the gap, Fig. 5(a) shows a sharp Majoranapeak at zero energy which broadens as γ increases29,30 andeventually disappears when γ → 1. Such a peak can be fittedwith the following Lorentzian function:
N (r,E) � N (�/2)2
E2 + (�/2)2 , (12)
where �/2 is the half width at half maximum and N is theheight at zero energy. The result of the fit is reported in Fig. 6:�/2 and N are plotted as functions of γ . Remarkably, �/2depends only very weakly on the position in the S-nanowirewhere the LDOS is calculated.
Interestingly, the Majorana peak is present also in the LDOSof the normal lead [Fig. 5(c)] as long as γ is not exactlyzero: The peak is still clearly distinguishable up to γ � 0.6.Moreover, singularities develop at energies corresponding to±Eg as γ tends to 1.
In the trivial phase, the LDOS in the S-nanowire at aposition close to the interface [Fig. 5(b)] presents a singlepair of peaks at γ = 0 (as compared to Fig. 4), since, beingsemi-infinite, the ends of the S-nanowire are sufficiently faraway to be decoupled. Such peaks quickly broaden as γ
increases and eventually merge into a single peak at γ � 0.3(a further increase of γ leads to the disappearing of the peak).As a result, as long as the coupling between the S-nanowireand the normal lead is not too strong, nontrivial and trivialphases can be distinguished from a measurement of the LDOSin the S-nanowire given a sufficiently large energy resolution(in the present case higher than 1% of the pairing �). On thecontrary, only a very weak double-peak structure is visible in
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the LDOS in the normal lead at a position close to the interfaceand for small values of γ [see Figs. 5(d)].
We mention that important clues on the topological phaseof a S-nanowire coupled to a normal electrode can be alsoretrieved using a different transport setup with respect to theone depicted in Fig. 2. Indeed, instead of considering thecurrent from the STM tip to the sample, it would, in principle,be possible to study directly transport through the NS junctionpresent in the nanowire. Even though a theoretical analysis ofthis configuration is beyond the scope of the present work, inAppendix B we report on the low-bias conductance of the NSjunction. We find that, in the tunneling limit, the approximatequantization of the conductance can be exploited to identify thetopological phase of the nanowire.29,31,32 We also remark that,even for transparent barriers, clear signatures of the presenceof Majorana fermions can be extracted from the transportproperties of the NS junction provided that a quantum pointcontact is present close to the interface.33
Let us now analyze how the LDOS changes when theS-nanowire is driven through a topological phase transition.34
In Fig. 7(a) we plot the topological invariant Q [calculatedusing Eq. (7)] and the number of open channels (Noc) asfunctions of the chemical potential μ for a fixed Zeeman fieldV = 0.3t . The S-nanowire goes through a transition, fromthe nontrivial (Q = −1) to the trivial (Q = +1) phase, atμ � 0.026t . Interestingly, the number of channels increasesfrom 1 to 2 at a much smaller value of the chemical potential
(a)
(b)
FIG. 7. (Color online) (a) Topological invariant Q (red solid line)and number of open transverse channels Noc (blue dash-dotted line) asfunctions of the chemical potential μ for a system with V/t = 0.3. Aphase transition occurs at μ/t � 0.026. (b) LDOS at a position insidea semi-infinite superconducting nanowire for different values of thechemical potential μ keeping V/t = 0.3 fixed. Results have beenoffset vertically for clarity, with an increasing value of the chemicalpotential from bottom (μ/t = 0) to top (μ/t = 0.05).
μ � 0.01t such that the nontrivial phase persists even inthe presence of two open channels. This observation is inapparent contradiction with the intuitive picture adopted inthe literature to explain the phase diagram of superconductingnanowires, that is, that a pair of Majorana fermions at theends of the wire is associated with each open channel and thatpairs of Majorana fermions on the same end can couple andform complex (Dirac) fermions. Accordingly, there shouldbe a single isolated Majorana fermion at each end of thenanowire whenever the number of open channels is odd. On theother hand, this intuitive picture should be treated with care,simply because one concept (presence of Majorana fermions)is related to a superconducting wire while the other (numberof open channels) to a normal one. Indeed, already in Ref. 17it was noticed that the system can be in the topologicallytrivial phase even when Noc is odd. Here, we are observingthe complementary situation in which the topological phasepersists when Noc is even. We believe that the underlyingexplanation is the same for both cases: the failure of theintuitive picture reported above to explain the whole phasediagram. As a consequence, we remark that the topologicalinvariant is not necessarily in a one-to-one correspondencewith the parity of the number of open channels. In Fig. 7(b)the LDOS as a function of energy is shown at a position insidethe S-nanowire (x = 50a, measured from the interface, andy = W/2 = 5a) when γ is close to zero. Different curvescorrespond to different values of the chemical potential, witha vertical offset proportional to μ. The Majorana peak splitsinto two Dirac-fermion peaks at finite energy as the chemicalpotential moves through the phase transition. Besides, wealso observe that the effective gap Eg initially decreasesfor increasing μ, then vanishes at the phase transition, andthereafter increases again. Similar results (concerning thedifferential tunneling conductance at one end of a nanowire)have been recently presented in Ref. 18.
C. SNS structure
Let us now consider two semi-infinite S-nanowires con-nected through a normal nanowire (N nanowire). For simplic-ity we assume transparent barriers at the interfaces and we set
�(i) ≡ �(x/a) =
⎧⎪⎨⎪⎩
�eiϕL x < −L/2,
0 |x| � L/2,
�eiϕR x > L/2;
(13)
that is, we allow for a finite phase difference �ϕ = ϕR −ϕL between the superconducting paring in the right andleft S-nanowires, L being the length of the N nanowire.Independently of the topological phase of the system, Andreevbound states (ABSs) appear as sharp peaks in the LDOS (seeFig. 8) and the ABS spectrum can thus be reconstructed fromthe energies at which such peaks occur. As we discuss atlength below, what differs between topologically trivial andnontrivial phases is the parity of the number of zero-energycrossings in the ABS spectrum, which is related to the presenceor absence of such a crossing at �ϕ = π (protected by fermionparity). This result is in agreement with similar calculationsperformed for a strictly one-dimensional system,13,35 for a
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FIG. 8. LDOS as a function of energy at a given position closeto the interface inside the left superconducting nanowire. Sharppeaks corresponding to Andreev bound states appear both withinthe effective gap Eg (� 0.007t) and inside the continuum. Theseresults refer to the following set of parameters: �ϕ = 2.0, μ/t = 0,V/t = 0.8, and L/a = 0.
two-dimensional SNS junction,36 and for a quantum spin-Hallinsulator sandwiched between superconducting leads.37
We start by considering the short-junction limit, L � ξ
(where ξ ∝ ta/Eg is the effective superconducting coherencelength), in which we have a single ABS for each open channel.L = 0 is a particular case in which the N nanowire is absentand we have a steplike jump from ϕL to ϕR in the phase of aS-nanowire. In Fig. 9 we show, for L = 0, the ABS energiesas a function of �ϕ for a topologically nontrivial [panel (a)for μ/t = 0 and V/t = 0.2] and trivial [panel (b) for μ/t =0.15 and V/t = 0.1] S-nanowires. In both cases we have asingle ABS (with its opposite energy counterpart). The maindifference between the two cases is that the number of zero-energy crossings is odd in the nontrivial phase while it is evenin the trivial case, in agreement with Ref. 13. Furthermore,even though the overall periodicity of the ABS spectrum is 2π
in both cases, the periodicity of a single branch of the ABSspectrum is 4π only for the nontrivial situation. In particular,the change of �ϕ by 2π at a fixed energy leads to the swappingof an ABS with its charge-conjugate state which has oppositefermion parity. In terms of Josephson current this leads tothe fractional Josephson effect9 and can be interpreted eitheras a 4π periodicity or as a two-valuedness of the Josephsoncurrent. For simplicity in the following we will address thetopologically nontrivial ABS spectrum as 4π periodic.
It is now interesting to compare the plot in Fig. 9(a)with the ABS spectrum relative to a short, one-dimensional,SNS Josephson junction with px + ipy superconducting orderparameter. An expression for the latter has been obtained,under the Andreev approximation (which assumes an orderparameter much smaller than the Fermi energy), in Ref. 38:
Eabs = ±�abs
√T cos (�ϕ/2) , (14)
where T is the transmission probability of the N regionand �abs is an effective order parameter for the ABS. Thesolid line in Fig. 9(a) shows the result of a best fit, withrespect to �abs, once T is set to 1. The agreement betweenthe numerical results (red open circles) and the fit is quite
0 π/2 π 3π/2 2π
Δϕ
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
Eabs/t
0 π 2π−0.002
0.000
0.002
0 π/2 π 3π/2 2π
Δϕ
−0.02
−0.01
0.00
0.01
0.02
Eabs/t
(a)
(b)
FIG. 9. (Color online) (a) Andreev bound-state energies (red opencircles) as a function of the phase difference �ϕ for topologicallynontrivial superconducting nanowires (μ/t = 0, V/t = 0.2, and γ =1). The solid line is a best fit using the expression in Eq. (14). (Inset)Same as in the main panel but for a nontransparent barrier betweenthe superconducting nanowires (γ = 0.1). (b) Same as in panel (a)but for topologically trivial superconducting nanowires (μ/t = 0.15and V/t = 0.1). Both panels refer to a SNS junction with L/a = 0.
good, despite the fact that the S-nanowire behaves effectivelyas a (px + ipy)-wave superconductor only in the limit oflarge Zeeman fields, with respect to � (in the present caseV = 0.2t and � = 0.1t).39 In the inset we show that, contraryto a nontopological SNS junction,40 the zero-energy crossingsurvives even when the barrier between the superconductorsis not transparent (γ = 0.1). In accordance with Eq. (14), theonly effect of a transparency T < 1 is to reduce the bandwidthof the ABS spectrum preserving the crossing at �ϕ = π .Indeed, with respect to the nontopological case discussed, forinstance, in Ref. 40, here the ABSs have different fermionparity; that is, they correspond to states with an even or oddnumber of fermions, respectively. Since fermion parity is aconserved quantity in this case, the two ABSs cannot couple(giving rise to the opening of a gap) even when the transparencyof the system is lowered.
Let us now address the case of larger Zeeman fields, where,however, the situation is complicated by the fact that byincreasing V also the number of open channels increases. InFig. 10(a) the ABS spectrum, as a function of �ϕ, is shownfor a topologically nontrivial S-nanowire with μ = 0 andV = 0.8t , where three transverse open channels are allowedin the absence of superconducting pairing. The spectrumpresents three ABS branches and an odd (three) numberof crossings at zero energy, as expected for the nontrivial
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0 π/2 π 3π/2 2π
Δϕ
−0.006
−0.004
−0.002
0.000
0.002
0.004
0.006E
abs/t
0.9π π 1.1π−0.002
0.000
0.002
0 π/2 π 3π/2 2π
Δϕ
−0.03
−0.02
−0.01
0.00
0.01
0.02
0.03
Eabs/t
(a)
(b)
FIG. 10. (Color online) (a) Andreev bound state energies (redopen circles) as a function of the phase difference �ϕ for topo-logically nontrivial superconducting nanowires when μ/t = 0 andV/t = 0.8. The solid lines are fits to the function in Eq. (14)with T = 1 for three different values of �abs for the three ABSs.The inset shows a magnification of the region close to �ϕ = π atsmall energies. (b) Same as in the panel (a) but for μ/t = 1.7 andV/t = 0.2. Both panels refer to a SNS junction with L/a = 0.
case. We fit separately the corresponding three ABS brancheswith Eq. (14), allowing three different values of �abs tobe interpreted as effective gaps for each open channel, onefor each branch. As shown in the plot, the fits (solid lines)apparently approximate quite well the numerical data (redopen circles). The agreement, however, is completely lostfor energies close to zero [see the inset of Fig. 10(a)]: (i)only one ABS branch crosses zero energy at �ϕ = π , (ii)the spectrum shows complicated avoided crossings whichshift the zero-energy crossing of the two remaining ABSbranches to �ϕ = π . As already discussed in Sec. III B, theMajorana modes, which would localize at a given end if thethree open channels were not coupled, strongly hybridize at�ϕ = π giving rise to a true Majorana mode plus a pair ofDirac excitations at finite energy. Moreover, we notice that theperiodicity of all three ABS branches is 4π and the peaks in theLDOS connected to the two ABSs with larger energy appear(see also Fig. 8) even above the bulk gap Eabs (�0.007t in thiscase).
The situation changes completely in the case of a smallerZeeman field (V/t = 0.2), still in the topologically nontrivialphase, while keeping the number of open channels equalto three (that is, by setting μ/t = 1.7). The correspondingspectrum, reported in Fig. 10(b), presents again an odd (three)number of zero-energy crossings, but shows gaplike features:
(a)
(b)
FIG. 11. (Color online) (a) Andreev bound-state energies (redopen circles) as a function of the phase difference �ϕ fortopologically nontrivial superconducting nanowires (V/t = 0.2 andμ/t = 0). (b) Same as in panel (a) but for topologically trivialsuperconducting nanowires (V/t = 0.05 and μ/t = 0). Both panelsrefer to the long-junction limit (L/a = 50).
No fitting with Eq. (14) is now possible. This is due to thefact that now the effective px + ipy description no longerholds since ABSs with both chiralities are present,11 so thatan interband s-wave pairing amplitude can mix them. In thiscase, one of the ABS branches has periodicity 4π while theremaining two have periodicity 2π . We still expect a fractionalJosephson effect.
If we increase the length of the N nanowire toward thelong-junction limit (L � ξ ), the number of ABSs increases butthe topological phase of the S-nanowire can still be detectedfrom the number of zero-energy crossings in the interval�ϕ ∈ [0,2π ].13,37 This is shown in Fig. 11, where the ABSspectrum is plotted for L/a = 50 (long-junction limit) whenthe S-nanowires are in the nontrivial phase [panel (a) forV/t = 0.2 and μ/t = 0] and in the trivial phase [panel (b)for V/t = 0.05 and μ/t = 0]. As in the short-junction limit,the periodicity of at least one branch of the ABS spectrum is4π only in the nontrivial case.
IV. CONCLUSIONS
In this paper we have considered a multiband semiconduct-ing nanowire subjected to spin-orbit coupling, superconduct-ing pairing, and a longitudinal Zeeman field. Depending on the
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GIBERTINI, TADDEI, POLINI, AND FAZIO PHYSICAL REVIEW B 85, 144525 (2012)
values of such parameters, the nanowire presents a nontrivialtopological phase in which a pair of Majorana modes, at anenergy equal to the chemical potential, are localized at itsends. We have first derived an analytic expression for the phaseboundaries of an infinitely long multiband nanowire. We havethen numerically calculated and analyzed the local density ofstates of such nanowires in the case when they are coupledto normal regions (such as electrodes or links) and we havecompared the topologically nontrivial and trivial phases indifferent situations. When the nanowire is coupled to a normalelectrode we have found that the peak in the local density ofstates at zero energy (with respect to the chemical potential),corresponding to the Majorana mode, broadens with increasingcoupling strength to the electrode, eventually disappearing fora transparent interface. Interestingly, for finite coupling thepeak is also present in the normal electrode, though beingof smaller amplitude and broadening more rapidly with thestrength of the coupling. In the trivial phase, and when thenanowire possesses two open channels in the absence ofsuperconducting pairing, a pair of peaks at finite energiesappears as due to the hybridization of the two Majorana modesthat would exist if the two channels of the nanowire werenot coupled. Such peaks broaden with increasing couplingstrength to the normal electrode, eventually merging forsufficiently large coupling. In the normal electrode only weakfeatures survive. From the analysis of the topological phasetransition, driven by varying the chemical potential at fixedZeeman field, we have found that the nanowire remains in thetopologically nontrivial phase even after the number of openchannels goes from one to two. This suggests that, contraryto the intuitive picture often referred to in the literature, theone-to-one correspondence between the topological invariantand the parity of the number of open channels is onlyapproximate and should be treated with care. We have thenconsidered the situation in which two semi-infinite nanowires(kept at different superconducting phases) are connectedthrough a normal link of length L. Independently of thetopological phase the density of states presents peaks dueto Andreev bound states whose position in energy dependson the superconducting phase difference �ϕ. While in thetrivial phase the number of zero-energy crossings is even, inthe topologically nontrivial phase this number is odd owingto the presence of a fermion-parity-protected crossing at�ϕ = π . This difference in the parity of the number ofzero-energy crossings reflects the presence of at least onebranch of Andreev-bound-state energy which is 4π periodic(instead of the usual 2π periodicity), leading to the so-calledfractional Josephson effect. This anomalous 4π periodicity ofthe Josephson current has been usually introduced in strictlyone-dimensional systems while we have checked that it sur-vives also in a multiband nanowire, in agreement with Ref. 41.
ACKNOWLEDGMENTS
We would like to acknowledge fruitful discussions withC.W.J. Beenakker and D. Rainis. This work has beensupported by the EU FP7 Programme under Grant Agree-ments No. 234970-NANOCTM, No. 248629-SOLID, No.233992-QNEMS, No. 238345-GEOMDISS, and No. 215368-SEMISPINNET.
APPENDIX A: DERIVATION OF THE PHASE DIAGRAM
We first need to determine the eigenvalues and eigenvectorsof HBdG(kx) for kx = 0 and π/a, when V = � = 0. Thedifference between kx = 0,π/a is just a shift ±2t in thechemical potential; that is,
HBdG(kx = 0,π/a) = H0 + [ε0 − (μ ± 2t)]τz. (A1)
Moreover, we are assuming � = 0, so that
H0 =(Hp 0
0 −σyH∗pσ
y
). (A2)
Thus, it suffices to consider the particle Hamiltonian Hp. For awire of width W/a = n the characteristic polynomial Ln(ε; α)of Hp can be defined recursively as
with L0(ε; α) = 1 and L1(ε; α) = ε. A formal solution to therecursive relation (A3) is given by
Ln(ε; α) = rn+1+ − rn+1
−√ε2 − 4(t2 + α2)
, (A4)
with
r± = ε ±√
ε2 − 4(t2 + α2)
2. (A5)
One can easily solve Ln(ε; α) = 0 and find the followingexpression for the energies εi
ελ = −2√
t2 + α2 cos
(λπ
n + 1
); λ = 1, . . . ,n. (A6)
For completeness, we mention that each eigenvalue is doublydegenerate and the corresponding eigenstates are characterizedby an amplitude ψ±
mσ on site m and spin σ given by
ψ±m,↑ = 1√
n + 1e∓imθ sin
(m
λπ
n + 1
)= ±ψ±
m,↓, (A7)
where the upper and lower signs refer to the two eigenstatesand tan θ = α/t . Moreover, we observe that even allowing fora finite Zeeman field V = 0, the eigenstates do not changesince we are assuming the Zeeman field and the SO couplingin the transverse direction to be both proportional to σx . Onthe other hand, if V = 0 the two eigenstates in Eq. (A7) areno longer degenerate, but split to energies εi ± V .
We now turn to the evaluation of the topological invariantQ in Eq. (7) when � = 0. We need to compute the Pfaffians:
Pf[HBdG(kx)σyτ y] with kx = 0,π/a. (A8)
We first introduce the 2n × 2n unitary matrix U of theeigenvectors (A7) of Hp and the associated 4n × 4n unitarymatrix UBdG,
UBdG =(
U 0
0 iσ yU ∗
), (A9)
which diagonalizes HBdG(0,π/a) when � = 0. Even if when� = 0 the matrix
D(kx) = U†BdGHBdG(kx)UBdG (A10)
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LOCAL DENSITY OF STATES IN METAL-TOPOLOGICAL . . . PHYSICAL REVIEW B 85, 144525 (2012)
is not diagonal, it is still useful to introduce it. Indeed, we canthen write
HBdG(kx)σyτ y = UBdGD(kx)(σyτ yUBdG)† (A11)
and, since by particle-hole symmetry we have that
σyτ yUBdG = U ∗BdGτ x, (A12)
the Pfaffian can be written as
Pf[HBdG(kx)σyτ y] = Pf[UBdGD(kx)τ xUT
BdG
]= det(UBdG)Pf[D(kx)τ x]
= (−1)nPf[D(kx)τ x]. (A13)
The product inside the Pfaffian reads
D(kx)τ x =(
E(kx) i�U †σyU ∗
−i�UT σyU −E(kx)
) (0 1
1 0
)
=(
i�U †σyU ∗ E(kx)
−E(kx) −i�UT σyU
), (A14)
where E(kx) is a diagonal matrix with entries given by{ελ + λV + ε0 − μ ∓ 2t}, λ = ±1. This matrix is indeedantisymmetric. One can easily see that, upon a reorderingof the rows and columns described by a real unitary matrixV , the matrix D(kx)τ x can be put into a block diagonal form,where each block is a 4 × 4 matrix involving the particle stateswith eigenvalues differing only by the sign of V and their holecounterparts. Namely, each block has the form⎛
⎜⎜⎜⎝0 −� ε + V 0
� 0 0 ε − V
−ε − V 0 0 �
0 −ε + V −� 0
⎞⎟⎟⎟⎠ , (A15)
where ε = ελ + ε0 − μ ∓ 2t . Thus, this means that
Pf[HBdG(kx)σyτ y]
= (−1)nPf[D(kx)τ x]
= (−1)n det(V)∏λ
[V 2 − �2 − (μ − ελ − ε0 ± 2t)2],
(A16)
where the upper and lower signs refer to kx = 0 and kx = π/a,respectively, and the matrixV is the same for both kx = 0,π/a.Thus, we finally have
Q = sgn{Pf[HBdG(0)σyτ y]Pf[HBdG(π/a)σyτ y]}=
∏λ,η=±1
sgn[�2 + (μ − ελ − ε0 + η2t)2 − V 2] (A17)
and consequently the phase boundaries are given by Eq. (8).We finally mention that analogous results have been foundin Ref. 42 in the case of spinless fermions in a p-wavesuperconducting nanowire.
APPENDIX B: TRANSPORT ACROSS A NS JUNCYION
In this Appendix we investigate another important toolto assess the topological phase of an S-nanowire coupledto a normal electrode: the low-bias conductance across aNS junction in the tunneling limit. In the limit of low
bias, transmission through the superconductor is completelysuppressed and the conductance of the NS junction can beexpressed in terms of the Andreev reflection matrix rhe (at thechemical potential)
G = 2e2
hTr[r†herhe] = 2e2
h
Noc∑m=1
Rm. (B1)
Here Rm are the eigenvalues of the Hermitian matrix r†herhe and
Noc is the number of open channels in the normal electrode.Owing to particle-hole symmetry, the Rm’s are either twofolddegenerate or equal to 0 or 1.33,43 The presence of a fullyAndreev-reflected mode (giving a quantized contribution to theconductance) is a signature of the existence of an uncoupledMajorana fermion at the Fermi energy.31,32 As a consequence,it is possible to write the conductance in the following form:33
G = 2e2
h
(1 − Q + 4
∑m
′Rm
), (B2)
where the primed sum is restricted to the degenerate Andreevreflection eigenvalues and Q is the topological invariant inEq. (7). In the limit of poorly transparent barriers (γ � 1),we expect that almost all modes are fully reflected (Rm ≈ 0,though never exactly zero)33,43 and thus
G ≈ 2e2
h(1 − Q). (B3)
As a consequence the low-bias conductance of the NS junctionin the tunneling limit gives an important information on thetopological phase of the S-nanowire.29,31,32 This result can beextended to almost transparent barriers if we include a ballisticquantum point contact close to the NS interface.33 In Fig. 12we show the conductance G (in units of 2e2/h) as a functionthe parameter γ controlling the transparency of the barrier. Inthe tunneling limit (γ � 1), we notice that the conductanceapproaches 0 for a topologically trivial S-nanowire (red dashedline, V/t = 0.2 and μ/t = 1.5) or 2e2/h for a topologicallynontrivial S-nanowire (blue solid line, V/t = 0.2 and μ/t =1.1), in agreement with Eq. (B3).
0.0 0.2 0.4 0.6 0.8 1.0γ
0
1
2
3
4
5
6
7
8
G[2
e2/h
]
FIG. 12. (Color online) Conductance (in units of 2e2/h) of the NSjunction as a function of the parameter γ controlling the transparencyof the barrier. The (blue) solid line refers to a S-nanowire in thetopologically nontrivial phase (V/t = 0.2 and μ/t = 1.1), while the(red) dashed line refers to a S-nanowire in the topologically trivialphase (V/t = 0.2 and μ/t = 1.5).
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GIBERTINI, TADDEI, POLINI, AND FAZIO PHYSICAL REVIEW B 85, 144525 (2012)
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